temperature-dependent absorption cross-sections of hcfc-142b
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Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy &Radiative Transfer
Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 364–371
0022-40
doi:10.1
� Cor
E-m
journal homepage: www.elsevier.com/locate/jqsrt
Temperature-dependent absorption cross-sections of HCFC-142b
Karine Le Bris a,�, Kimberly Strong b
a Department of Physics, Saint Francis Xavier University, P.O. Box 5000, Antigonish, NS, Canada B2G 2W5b Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7
a r t i c l e i n f o
Article history:
Received 31 July 2009
Received in revised form
2 October 2009
Accepted 7 October 2009
Keywords:
Hydrochlorofluorocarbons
R-142b
Cross-sections
Mid-infrared
FTIR
Gas phase
Density functional theory
Vibrational wave number
Band strength
73/$ - see front matter & 2009 Elsevier Ltd. A
016/j.jqsrt.2009.10.005
responding author. Tel.: þ1902 867 2392; fax
ail address: [email protected] (K. Le Bris).
a b s t r a c t
Following the recent detection of HCFC-142b (1-chloro-1,1-difluoroethane) from space,
laboratory infrared absorption cross-section spectra of this molecule in a pure vapour
phase have been recorded in the 65023500 cm�1 spectral region using Fourier
transform spectroscopy. The spectra have been recorded at a resolution of 0:02 cm�1
and a range of temperatures from 223 to 283 K. The resulting data show good agreement
with the harmonic frequencies and intensities calculated using density functional
theory as well as with the integrated absorption intensities of the spectral bands
available in the literature. The new cross-sections will allow more accurate retrieval of
atmospheric HCFC-142b concentrations using infrared spectroscopic techniques.
& 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Hydrochlorofluorocarbons (HCFCs) are temporary sub-stitutes for chlorofluorocarbons (CFCs). They were intro-duced after the phase-out of the latter by the MontrealProtocol and its subsequent amendments. Unlike CFCs,which are mainly destroyed by solar ultraviolet radiationin the stratosphere, the HCFCs, which contain one or morehydrogen atoms, can react with OH radicals in thetroposphere to create HF and CO2. Therefore, as lesschlorine is transported to the stratosphere, the ozonedepletion potentials of HCFCs are substantially weakerthan those of CFCs. However, because of their C–Cl andC–F bonds, these molecules still have large absorptioncross-sections in the atmospheric window regionð8212mmÞ, which give them strong global warmingpotentials.
ll rights reserved.
: þ1902 867 2414.
HCFC-142b (1-chloro-1,1-difluoroethane) is a colour-less gas at ambient pressure. It is mainly used as a che-mical intermediate to produce fluoropolymers, as ablowing agent for expanded polystyrene and as a compo-nent of refrigerant fluids. Because of its high vapourpressure and low vapour solubility, HCFC-142b partitionsmostly into the atmosphere. Today, this is the third mostabundant hydrochlorofluorocarbon after HCFC-22 andHCFC141-b. The atmospheric lifetime of HCFC-142b is17.9 years 724% [1]. Its ozone depletion potentialhas been estimated at 0.07 while its global warmingpotential is evaluated to be 23107810 for a horizon of 100years [1].
The volume mixing ratio of HCFC-142b has been risingsteadily in the atmosphere since the beginning of the1990s, exceeding 20 ppt in 2008 at 13–16 km with anaverage positive trend of more than 5% per year [2]. Theatmospheric concentration of HCFC-142b is now highenough to allow its detection from space [2,3]. However,the accuracy of the retrieval is limited by the lack ofspectroscopic knowledge of this molecule. Only threecross-section spectra corresponding to the temperatures
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K. Le Bris, K. Strong / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 364–371 365
253, 270 and 287 K are currently available in the HITRAN2008 database [4]. Measurements of temperature-depen-dent infrared cross-sections of HCFC-142b vapour at lowtemperature were reported in the 1990s [5–7] butdiscrepancies exist between these studies. No newlaboratory data at low temperature have been publishedon this molecule since 1995.
As a result, uncertainties on the spectroscopic para-meters of HCFC-142b are seen as one of the main sourcesof error for atmospheric retrievals [3]. New laboratorymeasurements are therefore crucial to enable accurateobservations of the spatial and temporal variation of thishydrochlorofluorocarbon in the atmosphere.
The purpose of this study is to provide new infrared(IR) cross-sections of HCFC-142b at relatively highresolution ð0:02 cm�1Þ and at a range of relevant atmo-spheric temperatures (from 223 to 283 K). The resultingexperimental data are compared to theoretical calcula-tions performed using density functional theory and topreviously published values.
2. Experimental setup
Experimental data are obtained using Fourier trans-form infrared (FTIR) absorption spectroscopy. The Fouriertransform spectrometer (FTS) is a Bomem DA8.002equipped with a KBr beamsplitter and operating with aGlobar source.
The gas sample (SynQuest Labs, purity 498%) iscontained in a stainless steel cell positioned betweenthe FTS and a liquid nitrogen-cooled mercury cadmiumtelluride (MCT) detector. A vacuum system equipped witha Varian Turbo-V V250 turbo molecular pump allowsthe cell to be evacuated to about 1� 10�6 Torr. The cellpressure is measured using 10 and 1000 Torr MKSbaratron pressure gauges simultaneously.
HCFC-142b presents very intense lines in the atmo-spheric window region. In order to avoid saturation effectswhile working at relevant pressures, a stainless steel cellwith an optical path of 3.17 cm was constructed. ZnSewindows were sealed to the cell with indium O-rings toprevent leakage at low temperature. The windows aremaintained in place by stainless steel flanges supportingteflon rings.
The cooling is achieved by a Neslab ULT-80 chillersending the coolant (Syltherm XLT) to a copper tubesurrounding the cell. The copper tube is soldered to thecell and covered by thermally conductive epoxy to providetemperature homogeneity. The cell temperature is mea-sured by a thermocouple directly inserted inside the cell.The temperature readout accuracy during experiments istypically 70:1 3C.
Many sources of errors and artefacts can affect an FTIRspectrum, such as spectral aliasing, dynamic alignmenterror, blackbody emission from the source aperture, non-linearity of the MCT detector in the mid-infrared, etc. Ther.m.s. angular deviation from optimum alignment is lessthan 10�6 rad in normal laboratory conditions (dataprovided by the manufacturer).
The blackbody emission from the source aperture comesfrom the warming of the iris by the light source. The warmannulus around the aperture acts as a second infrared sourceemitting off-axis thermal radiation. This leads to distortion ofthe signal shape and intensities as the aperture sizedecreases. To reduce this warm aperture artefact, a secondiris is inserted between the DA8 spectrometer and the MCTdetector at the focal point of the beam [8]. After thisinstallation, no variation of the cross-section signal with theaperture size has been observed.
The non-linearity of the MCT detector, correspondingto the non-linearity between the measured signal andthe photon flux, essentially affects the central fringe of theinterferogram, which leads to a zero-level offset in theabsorption spectrum. Correction of non-linearity is ap-plied on the raw interferograms before the phase correc-tion and the Fourier transformation. We adjusted theresponse of the detector to the light intensity by a curvewhose only unknown parameter is an empirically chosenDC output value [9]. This parameter is adjusted by aminimization of the transmission in the spectral regionbelow the cut-off wave number. The residual baselineoffset is subtracted from the spectra after phase correctionand Fourier transformation.
3. Data analysis
For each temperature, a series of spectra were recorded ata minimum of six different pressures between 2 and 12 Torr.Each pressure–temperature (P–T) spectrum is composed of atleast 200 unapodised scans. The resulting spectrum, for eachtemperature, is a composite spectrum from the P–T spectrumsets extrapolated to the zero-Torr limit. A 200-scan primarybaseline spectrum with an empty cell was recorded at eachtemperature. Secondary control baselines of at least 20 scanswere also taken before and after each sample measurementto account for the small intensity variations that can occurduring the long periods (typically several hours) of acquisi-tions. If necessary, the primary baseline is adjusted to thesecondary baselines by a polynomial regression prior to thedivision of each sample spectrum by the adjusted baselinespectrum.
The cross-section, sðnÞ, in cm2=molecule is calculatedfor each P–T set using the well-known Beer–Lambert law:
IðnÞ ¼ I0ðnÞ e�wðnÞ ð1Þ
with optical depth:
wðnÞ ¼ sðnÞ PT0
TP0NLL ð2Þ
where n is the wave number ðcm�1Þ; I0ðnÞ, the lightintensity passing through the empty cell (baseline);IðnÞ, the light intensity passing through the samplegas cell; NL is the Loschmidt’s constant ð2:6868�1019 molecules=cm3Þ; L, the length of the cell (cm); andP0 and T0, the standard conditions for pressure andtemperature.
To prevent saturation effects in an optically thickmedium while keeping a good signal-to-noise ratio atevery wave number, the points corresponding to opticallythick ðwðnÞ41:1Þ or optically thin ðwðnÞo0:1Þ conditions are
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K. Le Bris, K. Strong / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 364–371366
eliminated. This way, a linear behaviour is obtained forstrong absorption bands from the low-pressure measure-ments while the weak absorption features are representedby the high-pressure measurements.
The shapes of the ro-vibrational transition lines, aswell as the peaks of the Q-branches, are pressure-dependent due to collisional broadening. The cross-section for each wave number is obtained at the zero-Torr limit by the linear least-square fit of the experi-mental apparent cross-section of the remaining valuesð0:1owðnÞo1:1Þ versus the pressure.
The errors on optical path length, temperature readout,pressure, baseline drift, and sample purity have beenevaluated to account for less than 73%. The other sourcesof error in spectral measurements come from the residualMCT non-linearity, a possible residual baseline drift andthe errors induced by the data reduction. For each wavenumber, we calculated the standard deviation s betweenthe linear fit and the apparent cross-section values as afunction of pressure. The error for each wave number ischosen at the 95% confidence limit ð2sÞ.
4. Results
HCFC-142b is a near-prolate asymmetric top moleculebelonging to the Cs symmetry group (Fig. 1). As a conse-quence, it has 18 fundamental vibration modes (11 with A0
Fig. 1. Geometrical structure of HCFC-142b.
Fig. 2. Survey spectrum of HCFC-142
symmetry and 7 with A00 symmetry). Twelve of thosefundamental modes have active absorption bands in thespectral window region between 650 and 3500 cm�1.
All the bands present a sharp Q-branch surrounded by lessintense P- and R-branch structures (Fig. 2). No significantvariation of the integrated band strength with temperaturehas been observed from 223 to 283 K for any of the bands(Fig. 3). Therefore, we can confirm that the superimposed hotbands and combination bands of HCFC-142b do not alter thetotal integrated band intensities of strong fundamentalmodes in our range of temperatures.
The shape of the bands is strongly dependent on thetemperature. The Q-branches become sharper as the tem-perature decreases due to the change of relative populationsof the rotational states of the vibrational bands. This leads to asignificant increase in the cross-sections at the band centerswith decreasing temperature (Fig. 4).
The uncertainty on the integrated band strength isstable with temperature for all bands with the exceptionof the n8 band. The increased uncertainty on theintegrated strength of the n8 band as the temperaturedecreases is likely due to the intense sharpening of the n8
Q-branch which worsens the data reduction error on themain peak.
Weak features of magnitude lower than 10�20 cm2=
molecule, which can be attributed either to overtones andcombination band or sample impurity, are present atall temperatures in the spectral region between 1500and 2800 cm�1.
5. Data validation
5.1. Comparison with theoretical calculation of vibrational
modes and intensities
The unconstrained geometric optimizations and har-monic vibrational frequencies of HCFC-142b are calculatedusing density functional theory (DFT) with Gaussian 03 [10].
b in the mid-infrared at 263 K.
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6.8
7.2
0.8
0.08
220 290280270260250240230
0.160.4
0.6
2.2
2.4
6.4
ν6,ν5,ν14
ν4,ν3,ν13
ν1,ν2,ν12
ν7,ν15
ν8
Inte
grat
ed s
treng
th (c
m/m
olec
ule)
Temperature (K)
x10-17
Fig. 3. Variation of the integrated strength of the five main absorption bands of HCFC-142b in the mid-infrared.
966.0 966.5 967.0 967.50.0
0.5
1.0
1.5
2.0
283.15K
P
Cro
ss-s
ectio
n (c
m2 /
mol
ecul
e)
Wavenumber (cm-1)
Q223.15K
x1018
Fig. 4. Cross-section of the P- and Q-branches of the n15 band of HCFC-142b at seven different temperatures.
K. Le Bris, K. Strong / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 364–371 367
Previous ab initio calculations for HCFC-142b have beencarried out at the RHF/6-31* [11] and MP2/6-31G** [12]levels. The present DFT calculations are performed using theBecke’s three-parameter functionals for the exchange andthe Lee–Yang–Parr and Perdew–Wang non-local functionalsfor correlation (respectively, B3LYP and B3PW91). For eachfunctional set, two basis sets are used: the Pople’s typevalence triple-zeta basis supplemented by multiple polar-ization and diffuse functions ð6� 311þþð3df ;3pdÞÞ andthe augmented correlation-consistent polarized quadruple-zeta (aug-cc-pVQZ) basis.
We observe a very good agreement between thegeometries from the four levels of theory. The choice ofthe correlation functionals and the size of the basis setshave only a weak influence on the geometry. The
harmonic wave number positions present little sensitivityto basis sets, with an average difference below 0.2%, and tothe correlation functionals, with an average differencearound 0.7%. The variations of intensity with basis sets arealso weak in both cases, with the exception of the n10
band, corresponding to C–Cl stretching. It may be notedthat the inclusion of diffuse functions on hydrogen atomshas almost no influence on the geometry and harmonicfrequencies of HCFC-142b. However, the choice of correla-tion functionals does have a strong influence on theintensities, with an average difference between LYP andPW91 above 12% and a maximum difference between 25%and 30% for the n4 and n15 bands. Comparison withexperimental data is therefore essential to test the validityof the methods.
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Table 1Theoretical harmonic wave numbers and intensities of the fundamental modes of HCFC-142b.
Label Assignment Wave number Band strength
B3LYP B3PW91 B3LYP B3PW91
(a) (b) (a) (b) (a) (b) (a) (b)
A0
n1 CH3 sym. st. 3141.36 3136.41 3157.78 3152.34 0.06 0.06 0.05 0.05
n2 CH3 sym. st. 3063.13 3059.54 3071.15 3066.82 0.03 0.03 0.03 0.03
n3 CH3 sym. def. 1483.52 1483.83 1474.41 1474.57 0.01 0.01 0.01 0.01
n4 CH3 sym. def. 1421.65 1422.37 1412.26 1412.58 0.46 0.47 0.62 0.63
n5 CF2 sym. st., C-C st. 1217.65 1216.27 1231.51 1229.66 2.00 1.99 1.93 1.94
n6 C–Cl st., CF2 sym. def. 1129.57 1129.39 1128.24 1127.67 2.72 2.67 3.01 2.95
n7 CF2 sym. st. 892.16 889.67 903.03 900.49 2.10 2.14 1.83 1.88
n8 C–C–Cl def. 669.26 668.25 678.87 677.57 0.98 0.98 0.94 0.94
n9 CF2 scissor 538.74 538.32 543.53 543.01 0.22 0.22 0.21 0.21
n10 C–Cl st. 424.24 421.51 431.71 428.52 0.05 0.05 0.04 0.04
n11 CClF2�CH3 rock 301.07 299.86 301.40 299.99 0.02 0.02 0.02 0.02
A00
n12 CH2 asym. st. 3157.99 3153.42 3173.96 3169.17 0.03 0.03 0.02 0.02
n13 CH3 asym. def 1480.50 1480.27 1471.51 1470.95 0.06 0.06 0.06 0.06
n14 CF2 asym. st. 1185.37 1183.27 1194.77 1192.94 2.36 2.33 2.64 2.61
n15 CF2 asym. st. 963.01 961.31 968.74 967.87 1.20 1.2 0.93 0.97
n16 CF2 rock 430.36 429.96 431.12 430.89 0.00 0.00 0.00 0.00
n17 CClF22CH3 twist 330.49 329.79 332.15 331.29 0.01 0.01 0.01 0.01
n18 Torsion 236.96 235.49 239.07 238.32 0.00 0.00 0.00 0.00
Wave numbers are in cm�1, bands strengths are in 10�17 cm=molecule. (a) 6� 311þþ and (b) Aug-cc-pvqz. st:stretch, def: deformation.
K. Le Bris, K. Strong / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 364–371368
Table 1 presents the assignment, the calculatedharmonic wave number and the integrated strength ofthe fundamentals modes. The strongest bands correspondto C–C, C–Cl and C–F stretching vibrations. All of them fallin the 80021250 cm�1 atmospheric window region,which explains the high global warming potential of thismolecule.
The weak fundamental C–H stretching vibrations fall inthe 3000 cm�1 spectral region. The superimposed CH3
deformation vibrations lie in the 1400 cm�1 region. Onlythe n4 mode presents a medium intensity in this window.
The theoretical values of the harmonic frequencieshave been compared to the experimental data (Fig. 5). Theexperimental line center is chosen at the barycentre of theQ-band. All four levels of theory give excellent results afteran average linear scaling: nexp ¼ ð1:0670:01Þntheo� ð59:377:7Þ, with nexp and ntheo, the respective experimentaland theoretical harmonic wave number values in cm�1.The theoretical values for line intensity in the mid-IRspectral windows also fit relatively well the data afterlinear scaling with the experimental integrated absorptionstrength (see Fig. 6). We obtained an averageIexp ¼ ð1:0170:05ÞItheo þ ð0:2270:14Þ � 10�17 for B3LYPand Iexp ¼ ð1:107 0:02ÞItheo þ ð0:0870:05Þ � 10�17 forB3PW91, with Iexp and Itheo the respective experimentaland theoretical integrated band strengths in cm/molecule.However, B3PW91 gives the best correlation with an R2 of0.999 for both basis sets.
5.2. Comparison with published values
The integrated band intensities of the HCFC-142bcross-section spectra are compared with data available
in the literature (Table 2). The integrated intensities of thefour absorption bands n8, (n7, n15), (n6, n5, n14), and (n4, n3,n13) between 650 and 1500 cm�1 are reported. No pre-vious values for the n1, n2, n12 bands around 3000 cm�1
have been found in literature. At most temperatures,our integrated absorption band strengths show goodagreement with earlier measurements for a pure vapour[5,6] as well as for N2- broadened vapour [7,13]. Noconstant relative difference is observed with any of thedata sets, which indicates that there were no significantsystematic errors in the measurements.
The largest discrepancies between authors appears at233 K. A disparity close to 20% for the n6, n5, n14 absorptionbands exists between Newnham and Ballard [5] andCappellani and Restelli [7]. As previously mentioned, ourintegrated band strengths do not exhibit trends and ourvalues at 233 K, midway between the values of these twostudies, remain consistent with our results at othertemperatures.
6. Conclusions
Absorption cross-sections of HCFC-142b at a spectralresolution of 0:02 cm�1 have been recorded in the mid-infrared between 650 and 3500 cm�1 at seven differenttemperatures (223, 233, 243, 253, 263, 273, and 283 K).The main sources of errors in the measurements havebeen investigated and accounted for in the final results.The integrated intensities of the harmonic wave numberbands show good agreement with the data available in theliterature and with theoretical calculations using DFT. Theavailability of cross-section spectra at 10-K steps between223 and 283 K should, therefore, allow the reduction of
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750
750
1000
1000
1250
1250
1500
1500
3000
3000
3250
3250
B3LYP 6-311++B3LYP Aug-cc-pvqzB3PW91 6-311++
Theo
retic
al w
aven
umbe
r (cm
-1)
Experimental wavenumber (cm-1)
B3PW91 Aug-cc-pvqz
Fig. 5. Comparison between experimental and theoretical harmonic frequencies of HCFC-142b for the four levels of theory.
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
x10-17
x10-17
B3LYP 6-311++B3LYP Aug-cc-pvqzB3PW91 6-311++B3PW91 Aug-cc-pvqz
Theo
retic
al s
treng
th (c
m/m
olec
ule)
Experimental strength (cm/molecule)
Fig. 6. Comparison between experimental and theoretical band strengths of HCFC-142b for the four levels of theory.
Table 2
Comparison of the experimental infrared integrated band strengths ð�10�17 cm=moleculeÞ with published values when available.
Temperature Label Integrated band strength
This work Ref. [13]a Ref. [5]b Ref. [6]c Ref. [7]
293 K n8 0.74 0:7170:01 0.74
n7 , n15 2.53 2:4970:07 2.38
n6 , n5 , n14 7.06 7:3470:10 6.94
n4 , n3 , n13 0.64 0:5870:04 0.60
283 K n8 0:6970:02 0:7570:05
n7 , n15 2:4170:07 2:5870:14
n6 , n5 , n14 6:9970:19 7:1970:20
n4 , n3 , n13 0:5770:02 0:6170:09
n2 , n1 , n12 0:1370:01
273 K n8 0:7070:03 0.74 0.70 7 0.04 0:6970:05 0.75
n7 , n15 2:4270:07 2.50 2.56 7 0.07 2:5170:14 2.45
K. Le Bris, K. Strong / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 364–371 369
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Table 2 (continued )
Temperature Label Integrated band strength
This work Ref. [13]a Ref. [5]b Ref. [6]c Ref. [7]
n6 , n5 , n14 6:9170:19 7.05 7.06 7 0.13 7:1170:20 7.12
n4 , n3 , n13 0:6470:02 0.63 0:6170:03 0:6470:09 0.62
n2 , n1 , n12 0:1570:01
263 K n8 0:6870:06
n7 , n15 2:3570:06
n6 , n5 , n14 6:7470:18
n4 , n3 , n13 0:5970:02
n2 , n1 , n12 0:1470:01
253 K n8 0:6870:07 0:6970:03 0:6870:05
n7 , n15 2:3670:07 2:4170:08 2:4070:14
n6 , n5 , n14 6:7670:19 6:1670:43 6:9670:20
n4 , n3 , n13 0:6570:02 0:6570:05 0:7870:09
n2 , n1 , n12 0:1370:02
243 K n8 0:6770:11
n7 , n15 2:2970:07
n6 , n5 , n14 6:5870:18
n4 , n3 , n13 0:6470:02
n2 , n1 , n12 0:1370:02
233 K n8 0:6770:15 0:7170:01 0.75
n7 , n15 2:3570:07 2:3070:04 2.51
n6 , n5 , n14 6:9270:21 6:1970:02 7.38
n4 , n3 , n13 0:6170:02 0:6270:05 0.64
n2 , n1 , n12 0:1370:01
223 K n8 0:6470:15
n7 , n15 2:3070:08
n6 , n5 , n14 6:6770:19
n4 , n3 , n13 0:5870:02
n2 , n1 , n12 0:1270:01
213 K n8 0:7070:07
n7 , n15 2:4670:05
n6 , n5 , n14 6:8070:28
n4 , n3 , n13 0:6070:04
203 K n8 0:7970:09
n7 , n15 2:3470:26
n6 , n5 , n14 6:9070:10
n4 , n3 , n13 0:5870:03
a The first two temperature sets for Ref. [13] are at 297 and 277 K.b The first temperature set for Ref. [5] is at 296 K.c The first two temperature sets for Ref. [6] are at 287 and 270 K.
K. Le Bris, K. Strong / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 364–371370
uncertainties on HCFC-142b volume mixing ratio retrievalfrom space missions. The cross-sections are providedonline in supplementary data files and are also availablefrom the corresponding author.
Acknowledgements
This work was supported by the Canadian SpaceAgency (CSA) and the Natural Sciences and EngineeringResearch Council of Canada (NSERC). We thank Paul Chenfor technical support, and Prof. James R. Drummond andthe NSERC Industrial Research Chair in AtmosphericRemote Sounding from Space (sponsored by COMDEV,Bomem, Environment Canada, CSA, and NSERC) for the useof the Bomem DA8 Fourier transform spectrometer. Wealso thank Dr. Stella M.L. Melo from the CSA and Dr. Chris
Boone from the University of Waterloo for helpfuldiscussions.
Appendix A. Supplementary data
Supplementary data associated with this article can befound in the online version at doi:10.1016/j.jqsrt.2000.10.005.
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